Abstract
In this paper explicit formulas are given for the distribution functions and the moments of the local times of the Brownian motion, the reflecting Brownian motion, the Brownian meander, the Brownian bridge, the reflecting Brownian bridge and the Brownian excursion.
Highlights
We consider six stochastic processes, namely, the Brownian motion, the reflecting Brownian motion, the Brownian meander, the Brownian bridge, the reflecting Brownian bridge and the Brownian excursion
By letting the number of steps in the random walks tend to infinity we obtain the moments of local times of the processes considered
In each case the sequence of moments uniquely determines the distribution of the corresponding local time and the distribution function can be determined explicitly
Summary
We consider six stochastic processes, namely, the Brownian motion, the reflecting Brownian motion, the Brownian meander, the Brownian bridge, the reflecting Brownian bridge and the Brownian excursion. For each process we determine explicitly the distribution function and the moments of the local time. This paper is a sequel of the author’s papers [37], [41] and [42] in which more elaborate methods were used to find the distribution and the moments of the local time for the Brownian excursion, the Brownian meander, and the reflecting Brownian motion. We approximate each process by a suitably chosen random walk and determine the moments of the local time of the approximating random walk by making use of a conveniently chosen sequence of recurrent events. By letting the number of steps in the random walks tend to infinity we obtain the moments of local times of the processes considered. In a similar way we define rn(- a) (a 1, 2,..., n) as the number of subscripts r- 1,2,...,n for which (r-1 --a + 1 and (r- -a, that is, Vn(- a) is the number of transitions + -a 1--,- a in the first n steps
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